ELEG-212 Signal Processing
and Communications
READING ASSIGNMENTS
a This Lecture:
`Chapter 2, Section 2-6
a Other Reading:
`Appendix A: Complex Numbers
`Appendix B: MATLAB
`Next Lecture: start Chapter 3
LECTURE #3
Phasor Addition Theorem
a Lab 0 is this week – room change
`Sec. 10: eCALC II, 140 DuPont (Tuesday, 4:00)
`Sec. 11: eCALC I, 046 Colburn (Wednesday, 3:30)
`Secs. 12&13: eCALC II, 140 DuPont (Thursday, 3:30)
`Memory sticks/jump drives make saving work easy
ECE-212
Signal Processing First
LECTURE OBJECTIVES
Euler’s FORMULA
aPhasors = Complex Amplitude
aComplex Exponential
`Complex Numbers represent Sinusoids
`Real part is cosine
`Imaginary part is sine
`Magnitude is one
z (t ) = Xe jωt = ( Ae jϕ )e jωt
e jθ = cos(θ ) + j sin(θ )
Develop the ABSTRACTION:
Adding Sinusoids = Complex Addition
e jωt = cos(ω t ) + j sin(ω t )
PHASOR ADDITION THEOREM
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AVOID Trigonometry
Euler’s FORMULA
aAlgebra, even complex, is EASIER !!!
aCan you recall cos(θ1+θ2) ?
aComplex Exponential
aUse: real part of
ej(θ1+θ2)
`Real part is cosine
`Imaginary part is sine
`Magnitude is one
= cos(θ1+θ2)
e j (θ1 +θ 2 ) = e jθ1 e jθ 2
= (cosθ1 + j sin θ1 )(cosθ 2 + j sin θ 2 )
= (cosθ1 cosθ 2 − sin θ1 sin θ 2 ) + j (...)
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Real & Imaginary Part Plots
e jθ = cos(θ ) + j sin(θ )
e jωt = cos(ω t ) + j sin(ω t )
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COMPLEX EXPONENTIAL
e jω t = cos(ω t ) + j sin(ω t )
aInterpret this as a Rotating Vector
PHASE DIFFERENCE
`θ = ωt
`Angle changes vs. time
`ex: ω=20π rad/s
`Rotates 0.2π in 0.01 secs
= π/2
e jθ = cos(θ ) + j sin(θ )
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Rotating Phasor
Cos = REAL PART
Real Part of Euler’s
See Demo on CD-ROM
Chapter 2
cos(ω t) = ℜe{e jω t }
General Sinusoid
x(t) = Acos(ω t + ϕ )
So,
A cos(ω t + ϕ ) = ℜe{Ae
}
= ℜe{Ae ϕ e ω }
j (ω t +ϕ )
j
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COMPLEX AMPLITUDE
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j t
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WANT to ADD SINUSOIDS
General Sinusoid
x(t) = Acos(ω t + ϕ ) = ℜe{Ae jϕ e jω t }
aALL SINUSOIDS have SAME FREQUENCY
aHOW to GET {Amp,Phase} of RESULT ?
Sinusoid = REAL PART of (Aejφ)ejωt
x(t) = ℜe {Xe jω t }= ℜe{z(t)}
Complex AMPLITUDE = X
z(t) = Xe
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jω t
Signal Processing First
X = Ae
jϕ
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ADD SINUSOIDS
PHASOR ADDITION RULE
aSum Sinusoid has SAME Frequency
Get the new complex amplitude by complex addition
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ADD SINUSOIDS EXAMPLE
Phasor Addition Proof
tm1
tm2
tm3
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Convert Time-Shift to Phase
Phasor Add: Numerical
aMeasure peak times:
aConvert Polar to Cartesian
`tm1=-0.0194, tm2=-0.0556, tm3=-0.0394
`X1 = 0.5814 + j1.597
`X2 = -1.785 - j0.6498
`sum = X3
= -1.204 + j0.9476
aConvert to phase (T=0.1)
`φ1=-ωtm1 =-(2π/T)tm1 =70π/180,
`φ2= 200π/180
aAmplitudes
aConvert back to Polar
`A1=1.7, A2=1.9, A3=1.532
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ADD SINUSOIDS
`X3 = 1.532 at angle 141.79π/180
`This is the sumSignal
of Processing
the complex
amplitudes
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PHASOR EXAMPLES
X1
VECTOR
(PHASOR)
ADD
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X3
X2
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